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Metamath Proof Explorer

Theorem ge0mulcl

Description: The nonnegative reals are closed under multiplication. (Contributed by Mario Carneiro, 19-Jun-2014)

Ref Expression
Assertion ge0mulcl ( ( 𝐴 ∈ ( 0 [,) +∞ ) ∧ 𝐵 ∈ ( 0 [,) +∞ ) ) → ( 𝐴 · 𝐵 ) ∈ ( 0 [,) +∞ ) )

Proof

Step Hyp Ref Expression
1 elrege0 ⊢ ( 𝐴 ∈ ( 0 [,) +∞ ) ↔ ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) )
2 elrege0 ⊢ ( 𝐵 ∈ ( 0 [,) +∞ ) ↔ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) )
3 remulcl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 · 𝐵 ) ∈ ℝ )
4 3 ad2ant2r ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 · 𝐵 ) ∈ ℝ )
5 mulge0 ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → 0 ≤ ( 𝐴 · 𝐵 ) )
6 elrege0 ⊢ ( ( 𝐴 · 𝐵 ) ∈ ( 0 [,) +∞ ) ↔ ( ( 𝐴 · 𝐵 ) ∈ ℝ ∧ 0 ≤ ( 𝐴 · 𝐵 ) ) )
7 4 5 6 sylanbrc ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 · 𝐵 ) ∈ ( 0 [,) +∞ ) )
8 1 2 7 syl2anb ⊢ ( ( 𝐴 ∈ ( 0 [,) +∞ ) ∧ 𝐵 ∈ ( 0 [,) +∞ ) ) → ( 𝐴 · 𝐵 ) ∈ ( 0 [,) +∞ ) )